Answered

Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

At most, Alana can spend [tex]$\$[/tex] 40[tex]$ on carnival tickets. Ride tickets cost $[/tex]\[tex]$ 4$[/tex] each, and food tickets cost [tex]$\$[/tex] 2[tex]$ each. Alana buys at least 16 tickets. The system of inequalities represents the number of ride tickets, $[/tex]r[tex]$, and the number of food tickets, $[/tex]f$, she buys.

[tex]\[
\begin{cases}
r + f \geq 16 \\
4r + 2f \leq 40
\end{cases}
\][/tex]

What is the maximum number of ride tickets she can buy?

A. 4
B. 6
C. 10
D. 12

Sagot :

GinnyAnswer
To solve the problem and find the maximum number of ride tickets [tex]\( r \)[/tex] Alana can buy, let's follow a structured approach:

1. Define the variables:
- Let [tex]\( r \)[/tex] represent the number of ride tickets.
- Let [tex]\( f \)[/tex] represent the number of food tickets.

2. Set up the inequalities based on the problem description:
- Alana needs to buy at least a total of 16 tickets:
[tex]\[ r + f \geq 16 \][/tex]
- Alana has at most \[tex]$40 to spend where ride tickets cost \$[/tex]4 each and food tickets cost \$2 each:
[tex]\[ 4r + 2f \leq 40 \][/tex]

3. Simplify the inequalities:
- Simplify the second inequality by dividing through by 2:
[tex]\[ 2r + f \leq 20 \][/tex]

4. Consider the boundary conditions:
- To find the feasible region, let's convert the inequalities to equations:
[tex]\[ r + f = 16 \][/tex]
[tex]\[ 2r + f = 20 \][/tex]

5. Solve the system of equations:
- Set the equations equal to each other by substituting [tex]\( f \)[/tex] from the second equation into the first:
[tex]\[ f = 20 - 2r \][/tex]
Substitute [tex]\( f \)[/tex] into the first equation:
[tex]\[ r + (20 - 2r) = 16 \][/tex]
Simplify and solve for [tex]\( r \)[/tex]:
[tex]\[ r + 20 - 2r = 16 \][/tex]
[tex]\[ -r + 20 = 16 \][/tex]
[tex]\[ r = 4 \][/tex]

6. Verify the solution:
- Substitute [tex]\( r = 4 \)[/tex] back into one of the original inequalities to ensure it satisfies both conditions:
[tex]\[ r + f = 4 + f \geq 16 \quad \text{implies} \quad 4 + f = 16 \quad \text{so}\quad f = 12 \][/tex]
[tex]\[ 2r + f = 2 \times 4 + 12 = 8 + 12 = 20 \quad \text{(which is also less than 20)} \][/tex]

Thus, the maximum number of ride tickets Alana can buy is indeed [tex]\( 4 \)[/tex].

So, the answer is [tex]\(\boxed{4}\)[/tex].
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.